Zadanie 6
A field photographer packs 2 distinct lenses from 6 available and 3 distinct filters from 5 available. Two particular filters must not be packed together. How many different kits can be assembled?
Counting and generalized binomial expansions
Count selections and ordered arrangements, and expand binomials with rational powers within the required validity interval.
Tę umiejętność sprawdza 41 zadań wzorowanych na arkuszach IB z lat 2022–2026.
Zadania sprawdzające tę umiejętność: 41 Otwórz w wyszukiwarce z filtrami
Zadanie 9a
How many arrangements have no two silver objects adjacent?
Zadanie 9b
Require that no two silver objects are adjacent. How many arrangements additionally have silver objects at both ends?
Zadanie 11c
Show that the binomial expansion through x² is −7/4 + (17/16)x − (37/64)x2.
Zadanie 12e
The first two nonzero Maclaurin terms of x/(a + bx) match those in 12c, where a,b are rational. Find a,b.
Zadanie 1d
Show that , explaining the probability calculation for every coefficient.
Zadanie 7
In the binomial expansion about x = 0 of , where q is a positive rational number, the coefficient of x6 is 320. Find q.
Zadanie 8
A calibration network contains n sensors. Each unordered pair is scheduled to exchange three test messages. One sensor is removed after participating in exactly eight messages; all pairs among the remaining sensors complete their three messages. In total 701…
Zadanie 11a
Find the first four terms of the binomial expansion of F(x) in ascending powers of x.
Zadanie 11b
Expand R(x) in ascending powers of x through x2.
Zadanie 11c
The first three terms of these expansions agree. Prove that q = 2.
Zadanie 11d
Using p = 4, q = 2, the rational expression approximates F(x) near zero, with ∣x∣ < 1/6. (i) Choose a suitable x and use this approximation to estimate as a fraction of integers. (ii) Explain why the same approximation is less accurate for…
Zadanie 11a
How many different divisions of the samples into three groups are possible?
Zadanie 2
A field team has 12 specialists. Select an eight-person deployment group with three distinct roles: navigation lead, equipment lead and communications lead; the other five members have no assigned roles. How many selections are possible?
Zadanie 5a
(i) Show that a=3. (ii) Find b.
Zadanie 8a
How many run lists can be made with no restrictions?
Zadanie 8b
How many run lists keep all jobs of each test type consecutive?
Zadanie 8c
Three jobs of a single test type are chosen for a pilot; their order is irrelevant. How many selections are possible?
Zadanie 10a
(i) Show that the first four terms of (1 + 3x2)−1 are 1 − 3x2 + 9x4 − 27x6, for . (ii) Hence approximate its indefinite integral through the x⁷ term.
Zadanie 10b
(i) Expand (1 − 2x2)−1/2 through x⁴. (ii) Hence give the polynomial approximation to H(x) through x⁵.
Zadanie 7
Using mathematical induction and , prove for every integer n≥2.
Zadanie 9a
How many seating assignments satisfy this condition?
Zadanie 9b
Researchers C and D must also be nonadjacent, while A,B remain adjacent. Find the number of assignments.
Zadanie 9a
Express the number of assignments in terms of n.
Zadanie 9b
Two named engineers must now work in different crews. The number of permitted assignments becomes 8/15 of the original number. Find n.
Zadanie 7a
How many assignments are possible if the four probes must not form one adjacent block?
Zadanie 7b
Two additional distinct probes E,F are now installed. A,B,C,D must occupy an adjacent block of four. How many assignments of all six probes are possible?
Zadanie 1e
Conjecture A_n in terms of n.
Zadanie 9
Three distinct labels are chosen from {1, 2, …, 32}. Count the unordered selections whose label sum leaves remainder 1 on division by 3.
Zadanie 11e
Twelve trial outcomes are placed in the twelve numbered circular slots. If exactly three are successes, how many placements have all three success slots consecutive around the circle?
Zadanie 11f
Consider n≥4 trials with exactly three successes, placed in n numbered circular slots in trial order. (i) Give the number of placements in which all three success slots are consecutive around the circle. (ii) Find the greatest n for which the conditional…
Zadanie 11d
For experiment B, express P(B) in terms of y.
Zadanie 7a
Count linear arrangements when (i) there are no restrictions; (ii) all four metal pieces are consecutive.
Zadanie 7b
Count selections of five pieces containing at least three metal pieces.
Zadanie 7a
Count linear arrangements when (i) there are no restrictions; (ii) all four metal pieces are consecutive.
Zadanie 7b
Count selections of five pieces containing at least three metal pieces.
Zadanie 9a
Initially all five containers must occupy consecutive slots. Count the arrangements.
Zadanie 9b
They are rearranged with at least one empty slot between F and every other container. The other four need not be consecutive or separated from one another. Count the arrangements.
Zadanie 9a
How many codes have all digits distinct?
Zadanie 9b
How many have distinct digits in increasing order?
Zadanie 11a
Count the codes (i) if symbols may repeat; (ii) if all three symbols must differ.